On~the cardinality of compactifications of dyadic spaces

B. A. Efimov

Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 579-593

Résumé

It is shown that the supercardinality of a completely regular space $X$ equivalent to a dyadic space is equal to $\exp(\pi wX)$ provided $(\pi wX)^{\omega_0}=\pi wX$, where $\pi wX$ is the $\pi$-weight of $X$. In particular, it follows that the supercardinality of any countable dense subspace of a dyadic compactum of weight $\exp\omega_0$ is equal to $\exp\exp\omega_0$. This solves a problem raised by A. V. Arkhangel'skii on whether there exists a countable completely regular space whose every compactification has a cardinality larger than the continuum. Bibliography: 12 titles.